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Re: Preventing LegendreP from self-extracting in manipulations

• To: mathgroup at smc.vnet.net
• Subject: [mg58051] Re: Preventing LegendreP from self-extracting in manipulations
• From: Scott Hemphill <hemphill at hemphills.net>
• Date: Fri, 17 Jun 2005 05:18:56 -0400 (EDT)
• References: <d893ok$svp$1@smc.vnet.net>
• Reply-to: hemphill at alumni.caltech.edu
• Sender: owner-wri-mathgroup at wolfram.com

"Vladislav" <kazimir04 at yahoo.co.uk> writes:

> I have already tried to post the message, but it went lost. I repeat it
> as a new therad.
> 
> Can somebody help me with the following. I need present some functions
> (prolate spheroidal functions) in the basis of the Legendre
> polynomials. I.e.
> I have functions like
> 
> FF1 = 0.6 LegendreP[5, #1] + 0.7 LegendreP[6, #1] &
> FF2 = 0.3 LegendreP[5, #1] + 0.2 LegendreP[6, #1] &
> 
> I want to manipulate these functions and remain in the basis of prolate
> functions.
> For example I want to create a linear combination of functions, or
> something like this.
> 
> FF = .2FF1[#1] + .3FF2[#1] &
> 
> It works well from the point of view of finding the numerical result,
> but it do not give
> the presentaion of the function in the basis of the Legendre
> polynomials
> 
> I would like to have create a function which would give the result like
> 
> FFX = 0.9 LegendreP[5, #1] + 0.9 LegendreP[6, #1] &, so that I could
> see the presentaion
> of the function by typing FFX and obtaining  0.9 LegendreP[5, #1] + 0.9
> LegendreP[6, #1] &.
> In practice these functions contain much more terms and having the form
> like
>  0.9 LegendreP[5, #1] + 0.9 LegendreP[6, #1] & is very important. In
> the same way
> I would not like to have the explicit presentation as plolinomials,
> like -0.28125+
> 1.6875  w + 5.90625 w^2 + .. because of loss of accuracy for future
> results.

There are other ways to prevent loss of accuracy.  Instead of "0.9" as
a coefficient, use "9/10".  Or if fractional arithmetic becomes too
cumbersome, replace "0.9" with "N[9/10,32]" to specify extra precision.

Were the suggestions made by the other posters sufficient?

Scott
-- 
Scott Hemphill	hemphill at alumni.caltech.edu
"This isn't flying.  This is falling, with style."  -- Buzz Lightyear